Itegrals

[legacyofpiracy]

I'm not quite sure how to do this problem, would I be trying to take the antiderivative?

             x
If f(x)=   S    t^6 dt
             5

then f'(x)= ?
     f'(6)= ?

Any suggestions would be greatly appreciated

 

[pka]

\(\displaystyle \L
f(x) = \int_5^x {g(t)dt} \, \Rightarrow \quad f'(x) = g(x)\)

 

[legacyofpiracy]

Hah wonderful, now I realize how easy this problem actually was :roll: Thank you!

 

[Daniel_Feldman]

Yep. It's the Second Fundamental Theorem. Just be careful if you have something other than x. Then you would need to substitute (like you did here), and the multiply by the derivative of whatever your limit of integration is. For example,





\(\displaystyle \L
f(x) = \int_5^2x{g(t)dt} \, \Rightarrow \quad f'(x) = 2g(2x)\)




Edit: Due to my natural computer illiteracy, I can't get the above integral to output properly. Anyway, it should be from the definite integral, from 5 to 2x, of g(t)dt.